Congruences Satisfied by Apéry-Like Numbers

Chan, Heng Huat; Cooper, S. Barry; Sica, Francesco

doi:10.1142/s1793042110002879

Cited by 38 publications

(34 citation statements)

References 6 publications

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“…Recall from (3.10) that the sequence of central binomial coefficients is an LP function. Further armed with (3.5) as well as Corollary 3.5 and Lemma 3.6, the claimed Lucas congruences for the sequences 10 , s 7 follow from (3.6). It remains to consider the sequences (g), (δ), (ζ ) as well as (η) and s 18 .…”

Section: Lucas Congruencesmentioning

confidence: 71%

Divisibility properties of sporadic Apéry-like numbers

Malik

Straub

2016

Res. number theory

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In 1982, Gessel showed that the Apéry numbers associated to the irrationality of ζ(3) satisfy Lucas congruences. Our main result is to prove corresponding congruences for all known sporadic Apéry-like sequences. In several cases, we are able to employ approaches due to McIntosh, Samol-van Straten and Rowland-Yassawi to establish these congruences. However, for the sequences labeled s 18 and (η) we require a finer analysis.As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the Almkvist-Zudilin numbers are periodic modulo 8, a special property which they share with the Apéry numbers. We also investigate primes which do not divide any term of a given Apéry-like sequence.

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Section: Lucas Congruencesmentioning

confidence: 71%

Divisibility properties of sporadic Apéry-like numbers

Malik

Straub

2016

Res. number theory

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“…Some similar types of supercongruences on combinatorial numbers such as Almkvist-Zudilin numbers, Domb numbers and Apéry-like numbers have been studied by several authors, see for example, Amdeberhan and Tauraso [2], Chan, Cooper and Sica [5], Osburn and Sahu [16], and Osburn, Sahu and Straub [17]. In this paper, we aim to establish the same type of supercongruences on truncated 3 F 2 hypergeometric series.…”

Section: Introductionmentioning

confidence: 74%

Some supercongruences on truncated hypergeometric series

Liu

2017

Journal of Difference Equations and Applications

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Abstract. In 2003, Rodriguez-Villegas conjectured four supercongruences on the truncated 3 F 2 hypergeometric series for certain modular K3 surfaces, which were gradually proved by several authors. Motivated by some supercongruences on combinatorial numbers such as Apéry numbers and Domb numbers, we establish some new supercongruences on the truncated 3 F 2 hypergeometric series, which extend the four Rodriguez-Villegas supercongruences.

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“…In a recent paper [10], Chan, Cooper, and Sica investigated certain sequences of integers {f n } ∞ n=1 that satisfy relations similar to (1.1). The identification of a n as the coefficients of certain power series motivates them to obtain Apéry-like sequences {f n } ∞ n=1 satisfying congruences similar to (1.1).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%